Arrigo Cellina

On a classical problem of the calculus of variations without convexity assumptions

Abstract We show that the functional I ( x ) = ∫ 0 T g ( t , x ( t ) ) d t + ∫ 0 T h ( t , x ′ ( t ) ) d t attains a minimum under the condition that g be concave in x.

The existence question in the calculus of variations: a density result

We show the existence of a dense subset 0 of W(R) such that, for g in it, the problem rT rT minimum j g(x(t)) dt + h(x(t)) dt , x(O) = a, x(T) = b admits a solution for every lower semicontinuous h satisfying growth conditions

On evolution equations having monotonicities of opposite sign

(whenever meaningful), that applied to two solutions of (l), by the monotonicity of A and the minus sign on the right hand side, yields that their distance is nonincreasing. This reasoning allows us the construction of a Cauchy sequence of approximate solutions, converging to a solution. The existence of the right approximate solutions is supplied by the maxi- mality of A, that permits the use of the Yosida approximations. Hence existence is a result of completeness, of having the sign minus at the right hand side, and of maximality. The same conditions have allowed to prove existence for several classes of perturbations of (1) to i(f)E -Ax(t)+F(t,x(t)) in [4, 2, 9, 10, 14, 13, 11, 12, 15, 171. 71

Upper semicontinuous differential inclusions without convexity

We prove existence of solutions to the Cauchy problem for the differential inclusion x E A(x) , when A is cyclically monotone and upper semicontinuous.

A continuous version of Liapunov's convexity theorem

Abstract Given a continuous map s ↦ μs, from a compact metric space into the space of nonatomic measures on T, we show the existence of a family ( A α s ) α ∈ [ 0 , 1 ] , increasing in α and continuous in s, such that μ s ( A α s ) = α μ s ( T ) ( α ∈ [ 0 , 1 ] ) .

Approximate selections and fixed points for upper semicontinuous maps with decomposable values

We prove the existence of continuous approximate selections of upper semicontinuous maps from a separable locally compact metric space S into the decomposable subsets of L1 (T, Z). We then extend a fixed point theorem of Kakutani to upper semicontinuous maps with decomposable values.

Well posedness for differential inclusions on closed sets

For an initial value problem having uniqueness the well posedness, i.e., the continuous dependence on the initial condition, is expressed by saying that the map, assigning to the initial point the solution through it, is continuous. For problems lacking uniqueness, saying that any solution through a point can be embedded in a continuous, single valued, family of solutions depending on the initial point, can be considered as the natural extension of the well posedness. For a differential inclusion with Lipschitzean right-hand side, defined on an open set, several papers [3-6, 91 yield results on the existence of continuous selections from the map assigning to an initial point, or a parameter, the set of solutions to the corresponding Cauchy problem. The present note considers the same problem for a multifunction F defined on Rx K, where K is a closed subset of R, and satisfying a tangentiality condition, and proves an analogous result. Remark that the (generalized) successive approximations process that is the base of the construction, in the case under consideration requires at each step a projection over the (in general, non-convex) set K, since F is not defined outside K, and that this projection is not continuous. Moreover, the lack of an argument allowing the extension of a multivalued Lipschitzean map from a closed set to an open set containing it, prevents the possibility of exploiting the available techniques for the present case. As a side result we obtain the convergence of the sequence of generalized successive approximations for any initial function x0. As a corollary, we prove a result on the arcwise connectedness of the set of solutions and of